In this post I want to use Scholzian techniques to look at the relationship between modular forms and the etale cohomology of modular curves. Since everyone cares about modular forms, this exercise might be a good way of getting exposure to some of the latest ideas in p-adic Hodge theory. The basic reference for this post is Scholze’s paper “p-adic Hodge theory for rigid analytic varieties”, and most of the things stated without proof below can be found there (unless it’s false, in which case I made it up).

First a little background. Let be an extension of with perfect residue field , complete for a rank one valuation; any complete subfield of is fine. Let be a smooth rigid analytic space over . Often I really mean as an adic space over , for example whenever I start talking about topologies on , and always when I’m talking about proetale objects. The categories of qs rigid spaces and qs adic spaces locally of topologically finite type are canonically equivalent, so this isn’t too dangerous. In any case rigid spaces are more familiar.

We have the analytic site , the etale site , and the brand new spiffy proetale site , with their structure sheaves , , and . (WARNING: My is Scholze’s , and my is Scholze’s .) Opens in the proetale site are basically maps where is etale and is a directed inverse limit in which the morphisms are finite etale surjective. We have natural maps of sites and . The pullback is a mild operation: for any sheaf on , the natural map is an isomorphism. On a typical proetale open as above, we have for any abelian sheaf on .

Let be the de Rham complex of on the analytic site. Let be a pair consisting of a coherent locally free -module equipped with a separated and exhaustive decreasing filtration by local direct summands, and an integrable connection satisfying Griffiths transversality (i.e. carries into ). We get a filtration on the de Rham complex

by setting

so the associated graded pieces are given by

Define and . It’s a good sanity check to verify that if we take with for and for , then

In particular in this instance. In general we have the Hodge-de Rham spectral sequence

Note that for all intents and purposes, we can work interchangeably with ‘s on either or : writing and , the association is fully faithful and essentially surjective onto the obvious category of coherent locally free -modules with suitable filtration and connection, and we have compatibly with all structures, etc. We can also go from to and back again with no trouble, by the above remarks on .

On the other side of the universe we have locally constant sheaves on , and their pullbacks to . Recall that a lisse -sheaf on is an inverse system of sheaves of modules satisfying some natural finiteness and local constancy conditions. We define . Any lisse -sheaf gives rise to an honest sheaf of -modules on , and under some reasonable conditions we have , where the left side is defined by the usual inverse limit but the right side is honest-to-goodness sheaf cohomology!

Supposing the valuation on is discrete, there are certain canonically defined period sheaves , , and on . These are all sheaves of filtered rings. (As usual, the sheaves with pluses are subsheaves of the sheaves without pluses, and you remove the by inverting an element .) The sheaf is a relative version of Fontaine’s period ring : if we take and let be the directed system of all finite extension of in a fixed algebraic closure , then is a valid proetale open of , and . The sheaf is something muchstranger, with no good absolute analogue:

is a sheaf of -algebras, and .

comes equipped with a canonical integrable connection which satisfies Griffiths transversality, and the complex of sheaves is exact away from degree zero (and each filtered or graded piece is exact away from degree zero as well).

The subring of horizontal sections is canonically identified with .

Combining the second and third point, we get the all-important Poincaré lemma: the complex is a resolution of as sheaves on the proetale site. Since is a relative version of , which we all know is really the “correct” p-adic analogue of , this is pretty suggestive! It’s worth convincing yourself that there isn’t really any naive Poincaré lemma for the de Rham complex on a rigid analytic space: differential forms on rigid spaces just don’t like to get integrated without enlarging their domain, so already the de Rham complex of a closed affinoid polydisk isn’t exact. (This is one reason why rigid cohomology is a series of tubes.) Taking all of these behaviors together, it’s natural to think of as a mysterious bridge between the etale and analytic worlds.

As far as I know, Andreatta and Iovita deserve credit for realizing that Poincaré lemmas of this type exist and can be used to prove Fontaine’s comparison conjectures. It’s also interesting to note, in retrospect, the degree to which these constructions were {}“in the air”, in some cases for a surprisingly long time: for example, the ring we would now call and the zeroth graded piece of the Poincaré lemma both appear in a twenty-five year old paper of Hyodo, “On variation of Hodge-Tate structures”, with masquerading under the name “”.

It may be helpful to actually give the evaluations of these sheaves in a representative situation. Suppose is a reduced and irreducible smooth affinoid, say (or , it doesn’t matter). Let be the directed system of all finite etale extensions of contained in a fixed algebraic closure of . The object

is a perfectly good proetale open for . Set and , and let and be the p-adic completions of these rings. Then and . We have the usual theta map (whose kernel turns out to always be principal), and is exactly the -adic completion of . On the other hand, we can extend to a natural map

via the inclusion , and is the -adic completion of this tensor product; the connection is then gotten by extending the direct limit of the connctions . This completion process really ties and together in some inscrutable way, which is why I was using words like “strange” and “mysterious” earlier.

Definition. A lisse -sheaf on is de Rham if there exists as above such that

compatibly with the filtrations and connections on both sides. Such an is unique if it exists, in which case we say and are associated (or that is the associated -module of ).

Note that we can recover with its filtration and connection from via

Of course we could also take this as the definition of , for any , and then define to be de Rham if the natural map

is an isomorphism. This perhaps makes clearer the analogy with de Rham representations of . Also, just like in the case of Galois representations, the functor from to is certainly not fully faithful. Now we have the following result.

Theorem (Scholze).

a. If is smooth and proper, and is de Rham with associated -module , then the Hodge-de Rham spectral sequence for degenerates, and there is a canonical isomorphism

compatible with the filtrations and Galois actions. In particular, the etale cohomology is de Rham as a -representation, and we have a Hodge-Tate decomposition

b. If is the analytification of a smooth proper morphism of varieties, then is de Rham, with associated -module .

(In the numbering of Scholze’s paper, part a. is basically Theorem 8.4, and part b. follows from Theorems 8.8 and 9.3.)

This is already highly nontrivial in the case when and , in which case it becomes Fontaine’s de Rham comparison conjecture. The idea of the proof isn’t very difficult, and I’ll summarize it in the diagram

The middle isomorphism here is an immediate consequence of the Poincaré lemma. The lefthand arrow is an isomorphism by a surprisingly direct calculation, using the acyclicity of period sheaves on affinoid perfectoid opens in . The righthand arrow is also an isomorphism, by a calculation which seems (to my untutored eye) to be the least conceptual aspect of the proof. It seems tempting to reprove this step by studying the pushforward of under the projection : this should be a sort of Fréchet completion of , and might be interesting in its own right.

For the purposes of this post, let me assume the following mild generalization of these results. Suppose is smooth but not necessarily proper, with a lisse -sheaf on , and suppose is de Rham, with the associated -module with integrable connection. Say is compactifiable if we can find a smooth proper rigid space over together with an open immersion whose complement is a normal crossings divisor . Let us define

and

I will ASSUME that these groups are independent of the chosen compactification, and that they satisfy the obvious analogue of part a. of the theorem above. Probably the independence is straightforward (and written somewhere?), the de Rham comparison slightly less so.

Now fix prime to and an integer , and let and be the usual modular curves, considered as smooth rigid analytic spaces over . We have the universal elliptic curve . Let be the Tate module sheaf of ; this is a lisse -sheaf on . On the other hand, let be the first relative de Rham cohomology of with its natural filtration, so for and for , , and . Note that extends canonically to an invertible sheaf on denoted by the same letter, and (somewhat abusively) we have where denotes the cusps. It’s not hard to see by part b. of the theorem above that and are associated (on ).

Theorem. There is a canonical isomorphism

compatible with filtrations and Galois actions. Taking of this isomorphism gives a canonical Galois-equivariant short exact sequence

of -modules.

The first sentence is a trivial consequence of the assumption above and the fact that the notions of being de Rham and associated are preserved under functors like , but the second sentence requires an actual calculation. The reader who dislikes assumptions can amuse themselves by rewriting all of this in the context of quaternionic Shimura curves. Note that by the Kodaira-Spencer isomorphism (recalled below) we have

which by definition is the space of cusp forms of level . Likewise, by Serre duality and Kodaira-Spencer again we get

and is the space of all modular forms of level . Therefore taking the zeroth graded piece of the isomorphism in the theorem, we get the Hodge-Tate decomposition

This result was essentially obtained by Faltings twenty-seven years in his paper “Hodge-Tate structures and modular forms”. By the way, the reason I like this normalization is that if is a normalized cuspidal newform of level whose Hecke eigenvalues generate some finite extension , then

is exactly the Galois representation associated with , where is the ideal in the Hecke algebra generated by the operators .

We need to understand the filtration and grading of the logarithmic de Rham complex

The filtration vanishes in degrees and above. In degree we have

Next for we have

and the indicated map is nothing more or less than the Kodaira-Spencer isomorphism, so each of these graded pieces has trivial cohomology! Finally we have . Feeding all of this into the Hodge-de Rham spectral sequence, we find that the filtration on satisfies

This is enough to conclude the theorem.

It’s natural to ask: can we construct the map

directly, as a boundary map in the cohomology of some short exact sequence? Well, the Poincare lemma is a short exact sequence in this case; tensoring it with and recalling the definition of association, we get a filtered short exact sequence

of sheaves on . Now, is a subsheaf of of the rightmost term here, so we get a map

Taking of the above sequence and passing to its cohomology gives a connecting map